Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here: http://arxiv.org/pdf/1609.00588.pdf. There it is mentioned that no such $M$ exists for local Hopf algebras or algebras with radical cubed zero.

If such $M$ exist, then im curious about the following: Is there a nonprojective module $M$ over a finite dimensional local symmetric algebra with $Ext^{1}(M,M)=Ext^{2}(M,M)=0$? (If you can prove that no such $M$ exists, then you are probably famous since you proved the Tachikawa conjecture for this (large) class of algebras)